ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Nonlinearity
Abstract
We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural stringdiagrammatic extension of the approximately (realvalued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the socalled 'spider' associated with the computational basis, as well as a new arityN generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of nonlinear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZXcalculus. Consequently, Ncontrolled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical nonlinear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 DOI:
 10.48550/arXiv.1805.02175
 arXiv:
 arXiv:1805.02175
 Bibcode:
 2018arXiv180502175B
 Keywords:

 Quantum Physics
 EPrint:
 In Proceedings QPL 2018, arXiv:1901.09476